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Precision (Commons Math 3.2 API)
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org.apache.commons.math3.util</FONT>
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Class Precision</H2>
<PRE>
<A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">java.lang.Object</A>
  <IMG SRC="../../../../../resources/inherit.gif" ALT="extended by "><B>org.apache.commons.math3.util.Precision</B>
</PRE>
<HR>
<DL>
<DT><PRE>public class <B>Precision</B><DT>extends <A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</A></DL>
</PRE>

<P>
Utilities for comparing numbers.
<P>

<P>
<DL>
<DT><B>Since:</B></DT>
  <DD>3.0</DD>
<DT><B>Version:</B></DT>
  <DD>$Id: Precision.java 1422313 2012-12-15 18:53:41Z psteitz $</DD>
</DL>
<HR>

<P>
<!-- =========== FIELD SUMMARY =========== -->

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<B>Field Summary</B></FONT></TH>
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<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#EPSILON">EPSILON</A></B></CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 Largest double-precision floating-point number such that
 <code>1 + EPSILON</code> is numerically equal to 1.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#SAFE_MIN">SAFE_MIN</A></B></CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.</TD>
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<B>Method Summary</B></FONT></TH>
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<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;int</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double, double, double)">compareTo</A></B>(double&nbsp;x,
          double&nbsp;y,
          double&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compares two numbers given some amount of allowed error.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;int</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double, double, int)">compareTo</A></B>(double&nbsp;x,
          double&nbsp;y,
          int&nbsp;maxUlps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compares two numbers given some amount of allowed error.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double)">equals</A></B>(double&nbsp;x,
       double&nbsp;y)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true iff they are equal as defined by
 <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, 1)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)">equals</A></B>(double&nbsp;x,
       double&nbsp;y,
       double&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns <code>true</code> if there is no double value strictly between the
 arguments or the difference between them is within the range of allowed
 error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)">equals</A></B>(double&nbsp;x,
       double&nbsp;y,
       int&nbsp;maxUlps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are equal or within the range of allowed
 error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)">equals</A></B>(float&nbsp;x,
       float&nbsp;y)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true iff they are equal as defined by
 <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><CODE>equals(x, y, 1)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, float)">equals</A></B>(float&nbsp;x,
       float&nbsp;y,
       float&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are equal or within the range of allowed
 error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)">equals</A></B>(float&nbsp;x,
       float&nbsp;y,
       int&nbsp;maxUlps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are equal or within the range of allowed
 error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double)">equalsIncludingNaN</A></B>(double&nbsp;x,
                   double&nbsp;y)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or neither is NaN and they are
 equal as defined by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double)"><CODE>equals(x, y, 1)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double, double)">equalsIncludingNaN</A></B>(double&nbsp;x,
                   double&nbsp;y,
                   double&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or are equal or within the range
 of allowed error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double, int)">equalsIncludingNaN</A></B>(double&nbsp;x,
                   double&nbsp;y,
                   int&nbsp;maxUlps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or if they are equal as defined
 by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, maxUlps)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float)">equalsIncludingNaN</A></B>(float&nbsp;x,
                   float&nbsp;y)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or neither is NaN and they are
 equal as defined by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)"><CODE>equals(x, y, 1)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float, float)">equalsIncludingNaN</A></B>(float&nbsp;x,
                   float&nbsp;y,
                   float&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or are equal or within the range
 of allowed error (inclusive).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float, int)">equalsIncludingNaN</A></B>(float&nbsp;x,
                   float&nbsp;y,
                   int&nbsp;maxUlps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns true if both arguments are NaN or if they are equal as defined
 by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><CODE>equals(x, y, maxUlps)</CODE></A>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;boolean</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equalsWithRelativeTolerance(double, double, double)">equalsWithRelativeTolerance</A></B>(double&nbsp;x,
                            double&nbsp;y,
                            double&nbsp;eps)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns <code>true</code> if there is no double value strictly between the
 arguments or the reltaive difference between them is smaller or equal
 to the given tolerance.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#representableDelta(double, double)">representableDelta</A></B>(double&nbsp;x,
                   double&nbsp;originalDelta)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Computes a number <code>delta</code> close to <code>originalDelta</code> with
 the property that <code>
   x + delta - x
 </code>
 is exactly machine-representable.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#round(double, int)">round</A></B>(double&nbsp;x,
      int&nbsp;scale)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Rounds the given value to the specified number of decimal places.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#round(double, int, int)">round</A></B>(double&nbsp;x,
      int&nbsp;scale,
      int&nbsp;roundingMethod)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Rounds the given value to the specified number of decimal places.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;float</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#round(float, int)">round</A></B>(float&nbsp;x,
      int&nbsp;scale)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Rounds the given value to the specified number of decimal places.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static&nbsp;float</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/util/Precision.html#round(float, int, int)">round</A></B>(float&nbsp;x,
      int&nbsp;scale,
      int&nbsp;roundingMethod)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Rounds the given value to the specified number of decimal places.</TD>
</TR>
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<TH ALIGN="left"><B>Methods inherited from class java.lang.<A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</A></B></TH>
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<A NAME="EPSILON"><!-- --></A><H3>
EPSILON</H3>
<PRE>
public static final double <B>EPSILON</B></PRE>
<DL>
<DD><p>
 Largest double-precision floating-point number such that
 <code>1 + EPSILON</code> is numerically equal to 1. This value is an upper
 bound on the relative error due to rounding real numbers to double
 precision floating-point numbers.
 </p>
 <p>
 In IEEE 754 arithmetic, this is 2<sup>-53</sup>.
 </p>
<P>
<DL>
<DT><B>See Also:</B><DD><a href="http://en.wikipedia.org/wiki/Machine_epsilon">Machine epsilon</a></DL>
</DL>
<HR>

<A NAME="SAFE_MIN"><!-- --></A><H3>
SAFE_MIN</H3>
<PRE>
public static final double <B>SAFE_MIN</B></PRE>
<DL>
<DD>Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.
 <br/>
 In IEEE 754 arithmetic, this is also the smallest normalized
 number 2<sup>-1022</sup>.
<P>
<DL>
</DL>
</DL>

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<A NAME="compareTo(double, double, double)"><!-- --></A><H3>
compareTo</H3>
<PRE>
public static int <B>compareTo</B>(double&nbsp;x,
                            double&nbsp;y,
                            double&nbsp;eps)</PRE>
<DL>
<DD>Compares two numbers given some amount of allowed error.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - the first number<DD><CODE>y</CODE> - the second number<DD><CODE>eps</CODE> - the amount of error to allow when checking for equality
<DT><B>Returns:</B><DD><ul><li>0 if  <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><CODE>equals(x, y, eps)</CODE></A></li>
       <li>&lt; 0 if !<A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><CODE>equals(x, y, eps)</CODE></A> &amp;&amp; x &lt; y</li>
       <li>> 0 if !<A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><CODE>equals(x, y, eps)</CODE></A> &amp;&amp; x > y</li></ul></DL>
</DD>
</DL>
<HR>

<A NAME="compareTo(double, double, int)"><!-- --></A><H3>
compareTo</H3>
<PRE>
public static int <B>compareTo</B>(double&nbsp;x,
                            double&nbsp;y,
                            int&nbsp;maxUlps)</PRE>
<DL>
<DD>Compares two numbers given some amount of allowed error.
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent floating
 point numbers are considered equal.
 Adapted from <a
 href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
 Bruce Dawson</a>
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>maxUlps</CODE> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.
<DT><B>Returns:</B><DD><ul><li>0 if  <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, maxUlps)</CODE></A></li>
       <li>&lt; 0 if !<A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, maxUlps)</CODE></A> &amp;&amp; x &lt; y</li>
       <li>> 0 if !<A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, maxUlps)</CODE></A> &amp;&amp; x > y</li></ul></DL>
</DD>
</DL>
<HR>

<A NAME="equals(float, float)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(float&nbsp;x,
                             float&nbsp;y)</PRE>
<DL>
<DD>Returns true iff they are equal as defined by
 <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><CODE>equals(x, y, 1)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value
<DT><B>Returns:</B><DD><code>true</code> if the values are equal.</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(float, float)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(float&nbsp;x,
                                         float&nbsp;y)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or neither is NaN and they are
 equal as defined by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)"><CODE>equals(x, y, 1)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value
<DT><B>Returns:</B><DD><code>true</code> if the values are equal or both are NaN.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equals(float, float, float)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(float&nbsp;x,
                             float&nbsp;y,
                             float&nbsp;eps)</PRE>
<DL>
<DD>Returns true if both arguments are equal or within the range of allowed
 error (inclusive).
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>eps</CODE> - the amount of absolute error to allow.
<DT><B>Returns:</B><DD><code>true</code> if the values are equal or within range of each other.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(float, float, float)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(float&nbsp;x,
                                         float&nbsp;y,
                                         float&nbsp;eps)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or are equal or within the range
 of allowed error (inclusive).
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>eps</CODE> - the amount of absolute error to allow.
<DT><B>Returns:</B><DD><code>true</code> if the values are equal or within range of each other,
 or both are NaN.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equals(float, float, int)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(float&nbsp;x,
                             float&nbsp;y,
                             int&nbsp;maxUlps)</PRE>
<DL>
<DD>Returns true if both arguments are equal or within the range of allowed
 error (inclusive).
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent floating
 point numbers are considered equal.
 Adapted from <a
 href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
 Bruce Dawson</a>
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>maxUlps</CODE> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.
<DT><B>Returns:</B><DD><code>true</code> if there are fewer than <code>maxUlps</code> floating
 point values between <code>x</code> and <code>y</code>.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(float, float, int)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(float&nbsp;x,
                                         float&nbsp;y,
                                         int&nbsp;maxUlps)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or if they are equal as defined
 by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><CODE>equals(x, y, maxUlps)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>maxUlps</CODE> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.
<DT><B>Returns:</B><DD><code>true</code> if both arguments are NaN or if there are less than
 <code>maxUlps</code> floating point values between <code>x</code> and <code>y</code>.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equals(double, double)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(double&nbsp;x,
                             double&nbsp;y)</PRE>
<DL>
<DD>Returns true iff they are equal as defined by
 <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, 1)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value
<DT><B>Returns:</B><DD><code>true</code> if the values are equal.</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(double, double)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(double&nbsp;x,
                                         double&nbsp;y)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or neither is NaN and they are
 equal as defined by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double)"><CODE>equals(x, y, 1)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value
<DT><B>Returns:</B><DD><code>true</code> if the values are equal or both are NaN.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equals(double, double, double)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(double&nbsp;x,
                             double&nbsp;y,
                             double&nbsp;eps)</PRE>
<DL>
<DD>Returns <code>true</code> if there is no double value strictly between the
 arguments or the difference between them is within the range of allowed
 error (inclusive).
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - First value.<DD><CODE>y</CODE> - Second value.<DD><CODE>eps</CODE> - Amount of allowed absolute error.
<DT><B>Returns:</B><DD><code>true</code> if the values are two adjacent floating point
 numbers or they are within range of each other.</DL>
</DD>
</DL>
<HR>

<A NAME="equalsWithRelativeTolerance(double, double, double)"><!-- --></A><H3>
equalsWithRelativeTolerance</H3>
<PRE>
public static boolean <B>equalsWithRelativeTolerance</B>(double&nbsp;x,
                                                  double&nbsp;y,
                                                  double&nbsp;eps)</PRE>
<DL>
<DD>Returns <code>true</code> if there is no double value strictly between the
 arguments or the reltaive difference between them is smaller or equal
 to the given tolerance.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - First value.<DD><CODE>y</CODE> - Second value.<DD><CODE>eps</CODE> - Amount of allowed relative error.
<DT><B>Returns:</B><DD><code>true</code> if the values are two adjacent floating point
 numbers or they are within range of each other.<DT><B>Since:</B></DT>
  <DD>3.1</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(double, double, double)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(double&nbsp;x,
                                         double&nbsp;y,
                                         double&nbsp;eps)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or are equal or within the range
 of allowed error (inclusive).
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>eps</CODE> - the amount of absolute error to allow.
<DT><B>Returns:</B><DD><code>true</code> if the values are equal or within range of each other,
 or both are NaN.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="equals(double, double, int)"><!-- --></A><H3>
equals</H3>
<PRE>
public static boolean <B>equals</B>(double&nbsp;x,
                             double&nbsp;y,
                             int&nbsp;maxUlps)</PRE>
<DL>
<DD>Returns true if both arguments are equal or within the range of allowed
 error (inclusive).
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent floating
 point numbers are considered equal.
 Adapted from <a
 href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
 Bruce Dawson</a>
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>maxUlps</CODE> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.
<DT><B>Returns:</B><DD><code>true</code> if there are fewer than <code>maxUlps</code> floating
 point values between <code>x</code> and <code>y</code>.</DL>
</DD>
</DL>
<HR>

<A NAME="equalsIncludingNaN(double, double, int)"><!-- --></A><H3>
equalsIncludingNaN</H3>
<PRE>
public static boolean <B>equalsIncludingNaN</B>(double&nbsp;x,
                                         double&nbsp;y,
                                         int&nbsp;maxUlps)</PRE>
<DL>
<DD>Returns true if both arguments are NaN or if they are equal as defined
 by <A HREF="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><CODE>equals(x, y, maxUlps)</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - first value<DD><CODE>y</CODE> - second value<DD><CODE>maxUlps</CODE> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.
<DT><B>Returns:</B><DD><code>true</code> if both arguments are NaN or if there are less than
 <code>maxUlps</code> floating point values between <code>x</code> and <code>y</code>.<DT><B>Since:</B></DT>
  <DD>2.2</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="round(double, int)"><!-- --></A><H3>
round</H3>
<PRE>
public static double <B>round</B>(double&nbsp;x,
                           int&nbsp;scale)</PRE>
<DL>
<DD>Rounds the given value to the specified number of decimal places.
 The value is rounded using the <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true#ROUND_HALF_UP" title="class or interface in java.math"><CODE>BigDecimal.ROUND_HALF_UP</CODE></A> method.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - Value to round.<DD><CODE>scale</CODE> - Number of digits to the right of the decimal point.
<DT><B>Returns:</B><DD>the rounded value.<DT><B>Since:</B></DT>
  <DD>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="round(double, int, int)"><!-- --></A><H3>
round</H3>
<PRE>
public static double <B>round</B>(double&nbsp;x,
                           int&nbsp;scale,
                           int&nbsp;roundingMethod)</PRE>
<DL>
<DD>Rounds the given value to the specified number of decimal places.
 The value is rounded using the given method which is any method defined
 in <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><CODE>BigDecimal</CODE></A>.
 If <code>x</code> is infinite or <code>NaN</code>, then the value of <code>x</code> is
 returned unchanged, regardless of the other parameters.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - Value to round.<DD><CODE>scale</CODE> - Number of digits to the right of the decimal point.<DD><CODE>roundingMethod</CODE> - Rounding method as defined in <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><CODE>BigDecimal</CODE></A>.
<DT><B>Returns:</B><DD>the rounded value.
<DT><B>Throws:</B>
<DD><CODE><A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/ArithmeticException.html?is-external=true" title="class or interface in java.lang">ArithmeticException</A></CODE> - if <code>roundingMethod == ROUND_UNNECESSARY</code>
 and the specified scaling operation would require rounding.
<DD><CODE><A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</A></CODE> - if <code>roundingMethod</code> does not
 represent a valid rounding mode.<DT><B>Since:</B></DT>
  <DD>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="round(float, int)"><!-- --></A><H3>
round</H3>
<PRE>
public static float <B>round</B>(float&nbsp;x,
                          int&nbsp;scale)</PRE>
<DL>
<DD>Rounds the given value to the specified number of decimal places.
 The value is rounded using the <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true#ROUND_HALF_UP" title="class or interface in java.math"><CODE>BigDecimal.ROUND_HALF_UP</CODE></A> method.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - Value to round.<DD><CODE>scale</CODE> - Number of digits to the right of the decimal point.
<DT><B>Returns:</B><DD>the rounded value.<DT><B>Since:</B></DT>
  <DD>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="round(float, int, int)"><!-- --></A><H3>
round</H3>
<PRE>
public static float <B>round</B>(float&nbsp;x,
                          int&nbsp;scale,
                          int&nbsp;roundingMethod)
                   throws <A HREF="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</A>,
                          <A HREF="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</A></PRE>
<DL>
<DD>Rounds the given value to the specified number of decimal places.
 The value is rounded using the given method which is any method defined
 in <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><CODE>BigDecimal</CODE></A>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - Value to round.<DD><CODE>scale</CODE> - Number of digits to the right of the decimal point.<DD><CODE>roundingMethod</CODE> - Rounding method as defined in <A HREF="http://download.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><CODE>BigDecimal</CODE></A>.
<DT><B>Returns:</B><DD>the rounded value.
<DT><B>Throws:</B>
<DD><CODE><A HREF="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</A></CODE> - if an exact operation is required but result is not exact
<DD><CODE><A HREF="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</A></CODE> - if <code>roundingMethod</code> is not a valid rounding method.<DT><B>Since:</B></DT>
  <DD>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</DD>
</DL>
</DD>
</DL>
<HR>

<A NAME="representableDelta(double, double)"><!-- --></A><H3>
representableDelta</H3>
<PRE>
public static double <B>representableDelta</B>(double&nbsp;x,
                                        double&nbsp;originalDelta)</PRE>
<DL>
<DD>Computes a number <code>delta</code> close to <code>originalDelta</code> with
 the property that <pre><code>
   x + delta - x
 </code></pre>
 is exactly machine-representable.
 This is useful when computing numerical derivatives, in order to reduce
 roundoff errors.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>x</CODE> - Value.<DD><CODE>originalDelta</CODE> - Offset value.
<DT><B>Returns:</B><DD>a number <code>delta</code> so that <code>x + delta</code> and <code>x</code>
 differ by a representable floating number.</DL>
</DD>
</DL>
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